Theory of Numbers
Mathematics 356 — Fall 2009

Prof. Weibel



Catalogue Syllabus: Properties of the natural numbers, congruences, diophantine equations, prime numbers and elementary arithmetical functions.

Tentative Course Syllabus

Week Lecture dates Sections topics
19/3 (Thurs) 1.1-1.2 Numbers and Sequences, sums and products
29/8(T), 9/10 1.3-1.5; 2.1-2.2 Induction, Fibonacci numbers, Division Algorithm
Integer Representations and Operations
39/14, 9/17 2.3, 3.1-3.2 Complexity, Prime Number & their distribution
49/21, 9/24 3.2-3.4 GCDs and the Euclidean Algorithm
59/28, 10/1 3.5, 4.1-4.2 Fundamental Theorem, Congruences
710/5, 10/84.3, 4.5, 4.6 Chinese Remainder Thm, Factoring numbers (ρ method)
710/12, 10/15Review, Midterm Chapters 1-4
810/19, 10/226.1-6.2 Wilson's Theorem, Fermat's Little Theorem, pseudo-primes
910/26, 10/296.3, 7.1-7.2 Euler's φ function, Euler's Theorem, Sum and Number of Divisors
1011/2, 11/5 7.3-7.4 Perfect numbers, Möbius inversion
1111/9, 11/129.1-9.4 Primitive roots, Discrete Logarithms, Quadratic Residues
1211/16, 11/1910.1-10.2 El Gamal cryptosystem, Review
1311/23 (Mon) Midterm Chapters 6,7,9
1411/30, 12/311.1-11.2 Legendre symbols, Quadratic Reciprocity
1512/7, 12/1013.1-13.2 Pythagorean triples, Fermat's Last Theorem
1412/18 (Friday)Final Exam 8-11 AM

Handout on primitive roots
Due dateHomework Section/Problems
9/14/091.4 #4,16;   1.5 #5a,8,24,27
9/21/092.2 #14,16;   2.3 #4,14,16
9/24/093.1 #3,6,16(c),19,21;   3.2 #6(d),10(a),12,20(a)
10/1/093.3 #2f,10,22   3.4 #2bc,4bc,25   3.5 #2,6,10,16
10/8/094.1 #4,6(bcde),8,22,28(a);   4.2 #2(abc),8,10
10/15/094.3 #4(abc),8,12,17a,18;   4.5 #2a,4,10a,11a;   4.6 #2(b,d)
10/29/096.1 #3,4,10,12,26;   6.2 #1,2,7
11/5/096.3 #2,6,16;   7.1 #2,4(bc),12,20,30;   7.2 #2(de),10,12
11/12/097.3 #2,4(c),18;   7.4 #2(g),8,10,15,24,30,31;   9.1 #2(bd),6,8
11/19/099.2 #2(b,c),6,10;   9.3 #2,6(a),8(b);   9.4 #2(a),4,8
12/3/0911.1 #2(a,d), 4, 8, 20
12/10/0911.2 #1, 2, 4, 13(a,b,d);   13.1 #2, 4, 12


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Charles Weibel / Fall 2009